Math 2.0

Mechanics of particles and systems Definition

Rotating reference frame

Builds on Inertial frame of referenceAngular velocitySpecial orthogonal group

Definition

Fix an inertial frame with position coordinates x∈ℝ3x \in \R^3 and time tt in an interval II. A rotating reference frame with the same origin is given by a differentiable curve R ⁣:I→SO(3)R \colon I \to SO(3). Here SO(3)SO(3) is the special orthogonal group, and the coordinates X∈ℝ3X \in \R^3 of the frame satisfy x=R(t) X.x = R(t)\, X. The frame is genuinely rotating when its angular velocity ω(t)\omega(t), defined by R˙RTy=ω×y\dot R R^T y = \omega \times y, is not identically zero. It rotates uniformly when ω\omega is constant and nonzero.

Remarks

Example: a frame turning with constant angular speed Ω≠0\Omega \ne 0 about the zz-axis has R(t)R(t) equal to the rotation by the angle Ωt\Omega t about that axis, and ω=(0,0,Ω)\omega = (0, 0, \Omega). Non-example: a frame with R(t)=R0R(t) = R_0 constant is not rotating, since ω=0\omega = 0; it is an inertial frame with tilted axes. Differentiating x=RXx = R X gives x˙=RX˙+ω×x\dot x = R \dot X + \omega \times x: the inertial velocity is the velocity seen in the rotating frame, turned back to the inertial axes, plus the velocity ω×x\omega \times x due to the rotation. A frame whose origin also moves along a curve o(t)o(t) is given by x=o(t)+R(t)Xx = o(t) + R(t) X.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptRotating reference frame, definition, no evidence levelRotating reference fr…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cross product concept
  2. Derivative concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Identity matrix object
  6. Matrix definition
  7. Ordinary differential equation concept
  8. Vector space definition
  9. Velocity concept
  10. Classical mechanics concept
  11. Square matrix definition
  12. Transpose definition
  13. Linear map definition
  14. Inertial frame of reference concept
  15. Orthogonal matrix definition
  16. Multilinear map definition
  17. Orthogonal group object
  18. Alternating multilinear map definition
  19. Determinant definition
  20. Special orthogonal group object
  21. Angular velocity concept
  22. Rotating reference frame definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rotating_reference_frame
About this page
TagD72KEW2MFT
URIm2://D72KEW2MFT
Kinddefinition
WritingW0 stub
MSC70B05, 70A05
DefinesRotating reference frame
UsesInertial frame of reference, Angular velocity, Special orthogonal group
Changed3 November 2026

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